MNRAS 431, 1903–1913 (2013)
doi:10.1093/mnras/stt314
Advance Access publication 2013 March 14
The evolution of planetesimal swarms in self-gravitating
protoplanetary discs
Joe Walmswell,1‹ Cathie Clarke1 and Peter Cossins2
1 Institute
of Astronomy, Madingley Rd, Cambridge CB3 0HA, UK
of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK
2 Department
Accepted 2013 February 17. Received 2013 February 13; in original form 2012 December 20
ABSTRACT
We investigate the kinematic evolution of planetesimals in self-gravitating discs, combining
smoothed particle hydrodynamical (SPH) simulations of the disc gas with a gravitationally
coupled population of test particle planetesimals. We find that at radii of tens of au (which
is where we expect planetesimals to be possibly formed in such discs), the planetesimals’
eccentricities are rapidly pumped to values >0.1 within the time-scales for which the disc is in
the self-gravitating regime. The high resulting velocity dispersion and the lack of planetesimal
concentration in the spiral arms means that the collision time-scale is very long and that the
effect of those collisions that do occur is destructive rather than leading to further planetesimal
growth. We also use the SPH simulations to calibrate Monte Carlo dynamical experiments:
these can be used to evolve the system over long time-scales and can be compared with
analytical solutions of the diffusion equation in particle angular momentum space. We find
that if planetesimals are only formed in a belt at large radius, then there is significant scattering
of objects to small radii; nevertheless, the majority of planetesimals remain at large radii.
If planetesimals indeed form at early evolutionary stages, when the disc is strongly selfgravitating, then the results of this study constrain their spatial and kinematic distribution at
the end of the self-gravitating phase.
Key words: gravitation – protoplanetary discs.
1 I N T RO D U C T I O N
There are commonly discussed mechanisms by which gas giant
planets can form out of the gas and dust of the protoplanetary disc.
On the one hand, part of the disc may become sufficiently dense
to become gravitationally unstable and collapse (Boss 2000). Alternatively, in what has become to be known as the ‘core accretion’
scenario, the dust in the disc coagulates to form ever-larger objects, the planetesimals, eventually resulting in a solid core that
is large enough to initiate runaway accretion of a gaseous envelope (e.g. Pollack et al. 1996; Hubickyj, Bodenheimer & Lissauer
2005). In the latter scenario, planetesimal coagulation must occur
while the disc is still gas rich (in contrast to the case of the assembly of terrestrial planets, where the process may occur after
dispersal of the disc gas). Unsurprisingly, therefore, there is a large
literature devoted to planetesimal evolution in the presence of gas
(Adachi, Hayashi & Nakazawa 1976; Pollack et al. 1996; Tanaka
& Ward 2004). In most cases, the gas is treated as a laminar flow
although several authors (Laughlin, Bodenheimer & Adams 2004;
E-mail: [email protected]
Nelson & Papaloizou 2004; Nelson 2005; Ida, Guillot & Morbidelli
2008; Nelson & Gressel 2010) have considered instead the scenario where planetesimals are subject to stochastic torques arising
from fluctuations in a disc subject to turbulence generated by the
magneto-rotational instability. In the case of laminar discs, planetesimal eccentricity is (weakly) pumped by mutual gravitational
scattering (e.g. Safronov 1969) and (weakly) damped by gas drag,
so that the equilibrium distribution of eccentricities is peaked at low
values (∼0.01). This low value is important to continued planetesimal growth since it implies low relative velocities for planetesimal
encounters: this not only favours agglomerative (as opposed to destructive) outcomes (Benz & Asphaug 1999; Leinhardt, Stewart &
Schultz 2008) but also enhances the collision cross-section (∝ e−2 )
in the gravitationally focused regime.
All of the above studies treat the evolution of planetesimals in
non-self-gravitating discs. However, Rice et al. (2004) have argued
that planetesimals may be formed very early (in the first few 105 yr)
of a disc’s life when it is still strongly self-gravitating; at this stage,
spiral features in the disc provide pressure maxima in which dust is
concentrated through the action of gas drag. The enhanced collision
rates are favourable to grain growth, and Rice et al. (2006) have
argued that self-gravity in the dust phase may even promote the
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
1904
J. Walmswell, C. Clarke and P. Cossins
formation of km-scale structures (i.e. planetesimals). Observational
evidence for at least the early stages of grain growth during the selfgravitating phase is provided by the detection of 10 cm radiation
from HL Tau (Greaves et al. 2008), which implies that the growth
of grains to at least cm scales (from the sub-micron scales typifying
the interstellar medium) has already occurred in this young and
massive disc system.
If planetesimal formation indeed belongs to the earliest (selfgravitating) phase of disc evolution, then it is necessary also to trace
the evolution of planetesimals during the self-gravitating phase.
The dynamical evolution of planetesimals in this environment has a
number of implications for planet formation and for the collisional
production of dust in disc systems. For example, the relative importance of collisional growth of planetesimals in the self-gravitating
and non-self gravitating phase of the disc can be crudely assessed
(cf. Britsch, Clarke & Lodato 2008) by comparing the product of
the lifetime of each phase, the typical disc density and the (inverse)
square of the typical planetesimal velocity dispersion. The first two
terms roughly cancel (i.e. discs typically live an order of magnitude
longer in the non-self gravitating phase but with disc masses an
order of magnitude lower) so the relative importance of collisional
growth in the two regimes boils down to the relative values of the
velocity dispersion.
A pilot study of the dynamical evolution of planetesimals in
self-gravitating discs (Britsch et al. 2008) demonstrated that high
orbital eccentricities (e) are generated in such discs with particles
undergoing stochastic changes in their orbital elements as a result
of interaction with spiral features in the disc: high eccentricities
imply a high velocity dispersion (of the order of ev k where v K is the
local Keplerian velocity) if the particle trajectories are randomly
phased. A lower velocity dispersion would however apply if the
planetesimals instead demonstrated local velocity coherence: the
small number of particles modelled by Britsch et al. (2008) did not
permit exploration of this possibility however. For the same reason, it was not possible to discern the sign of any net migration
of the planetesimal swarm nor whether the main statistical effect
involved changes in orbital energy or angular momentum. All these
issues have implications for planetesimal growth during the selfgravitating phase and also for the retention of planetesimals against
loss through radial migration (Takeuchi, Clarke & Lin 2005) or
collisional grinding to small dust (Wyatt et al. 2007). Moreover,
evolution during the self-gravitating phase controls the spatial distribution and dynamical properties of any planetesimals that survive
this phase.
All of the above provides ample motivation for the present study
in which we explore the response of a large ensemble of planetesimals to the gravitational torques imposed by a self-gravitating gas
disc. We restrict ourselves to the regime where planetesimals respond as test particles to the imposed potential fluctuations. This
in practice restricts us to sizes >km scale (in order to be able to
neglect gas drag; Britsch et al. 2008) and <1000 km scale (in order
to be in the test particle regime where the effect of gravitational perturbations induced by the particles in the disc gas can be ignored;
Bate et al. 2003).
In Section 2, we present a numerical investigation [modelling
the disc with smoothed particle hydrodynamics (SPH)] that extends
the pilot study of Britsch et al. (2008) to a large ensemble of planetesimals. Our results (Section 3) imply the disc pumps particle
eccentricity with only modest changes in the semimajor axis and
this motivates the analytical modelling and Monte Carlo simulations
that we present in Section 4. This analytical/Monte Carlo approach
is able to reproduce the results of the SPH simulation over the rather
limited time frame that is possible in the latter case and also permits integration over long time-scales. In Section 5, we discuss the
results of these calculations in relation to the questions posed above
and summarize our conclusions.
2 T H E S I M U L AT I O N
2.1 The physics of self-gravitating discs
We model the gas disc as a self-gravitating disc which achieves a
situation of self-regulation: i.e. it settles to a marginally stable state
where the Toomre Q parameter (Toomre 1964) obeys
cs κ
∼ 1.
(1)
Q=
πG
Here, cs is the local sound speed, the gas surface density and κ
is the epicyclic frequency, which is ∼ (the Keplerian frequency)
for the low-mass discs considered here. In this state, the disc is in a
state of thermal equilibrium between the heating associated with the
gravitational instability and the imposed cooling. Here, we follow
Gammie (2001) and Lodato & Rice (2005) in parametrizing the
cooling in terms of a cooling time that is a fixed multiple (β) of the
local dynamical time (−1 ), i.e. we have tcool = β−1 and
u
Q− = −
,
(2)
tcool
where Q− and u are the cooling rate per unit mass and thermal
energy per unit mass, respectively.
The parameter β is a measure of overall cooling and the choice
of β determines the magnitude of the spiral density waves required
to maintain thermal equilibrium. By equating this cooling law with
the predicted heating rate per unit mass Q+ from the instability,
Cossins, Lodato & Clarke (2009) demonstrated that the fractional
density perturbation, a measure of relative spiral arm strength, is
proportional to β −1/2 ; the numerical simulations in the same study
confirmed this dependence and suggested that
δ
1
≈ √ .
β
(3)
This means that in our constant β simulations, we would expect
the density perturbation to be approximately constant across the disc
and to increase as β decreases. In practice, if β is less than about 4 in
a simulation, the magnitude of the perturbation becomes non-linear
and the disc fragments (Gammie 2001). (See e.g. Lodato & Clarke
2011; Meru & Bate 2011, 2012; Paardekooper, Baruteau & Meru
2011; Paardekooper 2012; Rice, Forgan & Armitage 2012 for an
ongoing debate as to whether fragmentation may also eventually
occur in well resolved simulations at significantly higher values of
β.)
2.2 Simulation parameters
We model the system as a point mass, mass M, orbited by 250 000
SPH gas particles and 50 000 test particles; for details of the code,
see Cossins et al. (2009) and references therein. The test particles are
assigned masses equal to 10−6 times the mass of a gas particle and
are subject only to gravitational forces. Gas particles are accreted
on to the point mass if they enter within a sink radius of 0.25 code
units and satisfy certain conditions (Bate, Bonnell & Price 1995);
the point mass itself is free to move. By the end of the simulation,
no more than 2 per cent of the gas particles are accreted.
Artificial viscosity is included, according to the standard SPH
formalism, with α SPH = 0.1 and β SPH = 0.2; for the parameters
Planetesimals in self-gravitating discs
employed, the ratio of SPH smoothing length to disc vertical scaleheight is about 2 throughout. The gas is modelled as a perfect
gas with γ = 5/3 and is subject to both compressive heating and
shock dissipation. The cooling law (equation 2) means that heat loss
depends only on the dimensionless parameter β, so cooling is scalefree. For the main simulation, we adopt β = 5 which corresponds
to rather large amplitude spiral perturbations (see equation 3); we
contrast these results with the lower amplitude case where β = 10.
We employ scaled units so that G = M = 1 and express time in
units of the dynamical time tdyn = −1 at R = 1. This time t always
refers to the time since the start of the simulation. We use these
dimensionless results because they can be easily scaled to real situations. Our standard model consists of a disc with total mass equal
to 0.1, distributed with surface density ∝R−3/2 over the radial range
1–25 and the planetesimals are distributed according to the same
radial distribution; we also investigate a similar set-up but with β =
10 and total mass of 0.5. We initialize the simulations with uniform
temperature so that Q = 2 at all radii in the disc. At each radius, we
distribute the particles in z in a Gaussian distribution representing
approximate hydrostatic equilibrium at the initial temperature. True
hydrostatic equilibrium is reached within a few local dynamical
times.
1905
Figure 1. Ring profile for R = 10 after five local periods and with initial
width of 1 at time t = 4000.
reality the disc mass would decline over such time-scales).1 On
this basis, we conclude that the majority of planetesimals would in
reality remain bound.
3.2 Ring spreading
3 S I M U L AT I O N R E S U LT S
The disc is initialized with Q = 2 throughout and is thus initially
gravitationally stable. It then cools on the local cooling time until
it attains Q1, initiating the gravitational instability and liberating
heat. By time t = 4000, the disc has definitely settled into a quasisteady state where spiral features form and dissolve on a roughly
dynamical time-scale and with Q1 for 5 < R < 25. (At R < 5, the
value of Q rises above unity since the disc is poorly resolved and
heating by artificial viscosity acting on the Keplerian velocity field
is sufficient to maintain the disc in a gravitationally stable state.)
Since this is a purely numerical artefact, we restrict our attention to
the regime R > 5.
3.1 Escapees
The first consideration is whether the test particles of the swarm
gained enough energy from their interactions with the disc to escape the system entirely. We assess this by calculating the energies
of the test particles (for this purpose we approximate the potential due to the disc by simply – for each particle – adding the
enclosed disc mass to the effective mass of the central object and
ignoring the contribution from external particles; although this is
not strictly correct in a disc system, this does not significantly
impair our ability to differentiate between bound and unbound
particles).
We find that (once the disc has settled into a steady gravoturbulent
state) around 200 particles (out of a total of 105 ) escape during the
subsequent 4000 time units. If we (somewhat arbitrarily) equate
code units for mass and radius with 1 M and 1 au then this implies
an e-folding time-scale of 3 × 105 yr (i.e. of the order of the selfgravitating lifetime of the disc). This e-folding time-scale is likely to
be an underestimate for two reasons: the low value of β employed in
the simulations (which implies much more vigorous spiral activity
than would be expected given the relatively long cooling timescales in realistic protostellar discs; Clarke 2009 and Stamatellos
et al. 2007) and the assumption of constant disc mass (whereas in
Once the system has settled into the equilibrium state (i.e. after t =
3–4000), we identify rings of particles over a restricted radial range
and analyse their spatial evolution. Fig. 1 illustrates the rather rapid
spreading after five local orbital periods of a ring of initial width 1,
initially located at R = 10 at t = 4000. We find that the fractional
change in the centroid of the distribution and the change in the width
of the distribution (normalized to the initial radius) is independent
of the radius of the ring selected as long as the comparisons are
conducted after the same number of local dynamical time-scales.
This is as expected, given that (for the constant β that we impose)
the fractional amplitude of the gravitational perturbations should be
independent of radius. Note that although the mode of the particle
radii has moved in slightly, the mean has moved out. Fig. 2 illustrates
the evolution of the fractional standard deviation of a ring at R =
20 selected at t = 3500 and illustrates the continued spreading of
the ring over many local orbital times.
We find that the rings spread in response to stochastic interaction
with the disc and also that there is a small outward shift in the
particle centroid for each ring. We can understand this behaviour
by considering the evolution of the energy distribution of particles
in a selected ring (Fig. 3): the finite width of the initial distribution
just reflects the finite width of the ring. It is notable that there is
little evolution in the distribution of particle energies over the time
period of t = 4000–7500, whereas ring spreading is still significant
over this period (Fig. 2). Although this spreading can be partially
attributed to phase mixing (given the finite eccentricities of the particles selected), this is not the only process at work, as is illustrated
by Fig. 5, which shows the secular increase of the particle eccentricities. This suggests that as an ensemble, the particles are undergoing
little evolution in energy but are diffusing in angular momentum.
This implies a growth of particle eccentricity which we quantify
below for the entire disc. [It should be stressed that the constancy
1 We found an approximately three times higher escape rate in the case of our
massive disc simulation (with disc mass five times higher than the standard
case), which demonstrates that – as expected – the rate of test particle escape
would drop as the disc mass declines.
1906
J. Walmswell, C. Clarke and P. Cossins
Figure 2. Evolution of the ring standard deviation at R = 20. The ring is
selected at t = 3500, when the disc is in the quasi-steady state, and then
followed for the rest of the simulation.
Figure 4. Eccentricity evolution of the standard simulation.
Figure 5. Evolution of the eccentricity means for the two simulations.
Figure 3. Evolution of the energy distribution of the R = 10 ring, selected
at t = 4000.
of energy – and hence semimajor axis – is a property of the local
ensemble; as was noted in Britsch et al. (2008), individual particles undergo stochastic interactions in which they undergo modest
changes in energy.]
3.3 Eccentricity growth
Fig. 4 depicts the eccentricity distribution for the entire disc ensemble over the duration of the simulation and confirms the evolution
towards higher eccentricity, as would be expected if the spatial
spreading largely reflects the growth of particle eccentricity at fixed
semimajor axis. Note that in the period before the disc reaches the
quasi-steady state (i.e. between t = 0 and t = 3−4000), the eccentricity distribution is not being pumped by the recurrent spiral
features. There does not appear to be a sharp transition between the
two regimes though.
In Fig. 5, we compare the evolution of the mean eccentricity
between the standard simulations and the high β case (in which β
differs by a factor of 2). We see that the time-scale to attain a given
mean eccentricity is roughly doubled in the higher β case. This
is consistent with a picture in which the particles are responding
diffusively to gravitational interactions with the disc: the diffusion
time-scale scales with the inverse of the diffusion coefficient and
thus with the inverse square of the amplitude of the perturbations.
Hence (from equation 3), we expect the time-scale for eccentricity
growth to scale linearly with β, as is consistent with Fig. 5. We note
that the mean eccentricity increases smoothly after t = 500 and
that, as with the distribution as a whole, we cannot identify a sharp
transition when the quasi-steady state is reached. This means that
we can consider the distribution to be a reasonable proxy for how it
would appear had the disc been in the quasi-steady state throughout,
a fact we use in Section 4 to scale our analytical model and Monte
Carlo simulations.
3.4 The velocity dispersion
For a swarm of particles with randomly orientated orbits, the velocity dispersion is approximately equal to the product of the eccentricity and the local Keplerian velocity (Britsch et al. 2008). The
strong growth in particle eccentricity would thus raise the velocity dispersion of the planetesimals with consequences for particle
growth that we discuss below. However, this would be an overestimate of the velocity dispersion if there was a degree of local
velocity coherence within the particle swarm. We investigate this in
Fig. 6 by comparing the evolution of the local velocity component
(σ ) and its three orthogonal components (computed within a patch
Planetesimals in self-gravitating discs
1907
during the self-gravitating phase. Once the disc enters the non-self
gravitating phase, and gas drag reduces the equilibrium eccentricity
level to around 0.01 (Kokubo & Ida 2000), the collision rate rises
by around a factor of 100 and at this stage either collisional growth
and/or dust production may become important.
3.5 Planetesimal concentration in the arms
Figure 6. Evolution of the velocity dispersion (σ ) and its components in
the radial (σR ), azimuthal (σ φ ) and vertical (σ z ) directions, compared with
the prediction for randomly orientated orbits (σp ).
of disc containing ∼100 particles) with the product of the instantaneous mean eccentricity and the local Keplerian velocity (σ p =
evk ). The figure illustrates that ev k is indeed a good measure of the
local velocity dispersion, as is expected in the case of randomly
phased elliptical orbits. In addition, the ratio between the radial and
azimuthal dispersions is maintained at about 3:2. One can show
from the epicyclic approximation for a cylindrical potential that
the expected ratio between the dispersions should be σφ2 = 0.5σR2
√
(Binney & Tremaine 2008), i.e. that σφ = σR / 2 or 3σφ ≈ 2σR .
The growth of the velocity dispersion has profound consequences. First, a high velocity dispersion suppresses the role of
gravitational focusing in facilitating collisions (Britsch et al. 2008).
In the limit that the disc scaleheight exceeds the mean separation between planetesimals the time-scale for physical collisions between
planetesimals of mass M, radius R and density ρ is
tgrow ∼
ρ 2/3 M 1/3
,
p (1 + 4 GM/(Rσ 2 ))
(4)
where p is the planetesimal surface density and σ is the planetesimal velocity dispersion. The σ dependence means that collisions
are highly disfavoured. At 30 au from a solar mass star and using
e = 0.2, the velocity dispersion is around 1 km s−1 . Using the above
formula for 100 km planetesimals, the collision time-scale is of the
order of 109 yr! Smaller planetesimals would give even longer timescales. Since planetesimal growth cannot proceed without physical
collisions, this rules out significant planetesimal growth during the
self-gravitating phase.
In addition – even when collisions do occur – the collisional
velocity is correspondingly high. The velocity required to shatter
planetesimals by collision is a few 100 m s−1 (Leinhardt et al. 2008),
meaning that collisions would tend to result in disruption rather
than in growth. We can then use the above collision probabilities to
estimate a rough upper limit on the amount of small dust (i.e. in the
observable regime of <1 mm) that could be generated by disruptive
collisions between planetesimals. On the optimistic assumption that
every planetesimal–planetesimal collision results in all its mass
being liberated as dust, the fraction of mass in planetesimals that
can be liberated is simply the ratio of the self-gravitating lifetime to
the collision time, i.e. ∼10−4 . We therefore conclude that the role
of planetesimal collisions in re-supplying small dust is insignificant
The above estimates for planetesimal collision frequencies neglect
any possible concentration of particles in spiral features. For the
values of β employed in the simulations, the gas surface density
varies by no more than a factor of 2 between the arms and the interarm regions (as expected, since we have modelled a regime where
the disc is close to – but not at – the fragmentation boundary). If
the planetesimals simply followed the gas, such an enhancement
would have only a minor effect on planetesimal collision probabilities, even in the unlikely event that planetesimals spent all their
time in long-lived regions of density enhancement. In fact, we find
that – instead of being preferentially concentrated in the arms –
the planetesimal surface density varies by no more than ∼20 per
cent around the orbit in the standard simulation, i.e. with amplitude
much less than that of the gas.2
Britsch et al. (2008) suggested that there were possible hints that
the concentration of particles in the arms was more efficient in the
case of more massive discs. In their simulation with a disc mass of
half that of the point mass they identified epochs at which some test
particles would settle into orbits that corotated with the spiral modes
and conserved a Jacobi constant. We have analysed our ‘massive
disc’ simulation and however find that, as in the standard simulation,
there is no discernible influence on the planetesimal distribution:
there is no significant concentration of planetesimals in the arms.
3.6 A truncated planetesimal distribution
We finally consider the case that the planetesimal distribution is
truncated, i.e. that it does not initially extend within a certain inner
radius, Rin , and study the extent to which planetesimals are scattered
into regions within Rin . This choice is motivated by the suggestion
of Clarke & Lodato (2009) that planetesimals are only likely to form
in self-gravitating discs at large radius (beyond a few tens of au). We
also note that the simulations of Gibbons, Rice & Mamatsashvili
(2012) suggest that the mechanism is probably restricted to radii
greater than 20 au. At such radii, the cooling time is relatively short
(corresponding to β < 10; Clarke 2009) and the amplitude of spiral
disturbances is large enough for rapid concentration of dust in spiral
structures [i.e. on time-scales shorter than the (roughly dynamical)
lifetimes of spiral features].
We do not need to conduct new simulations for this scenario but
– since the planetesimals are non-interacting – simply tag planetesimals that are located at radius R > 10 at t = 5000 and follow the
evolution of their density distribution. From Fig. 7, one can see
that the distribution rapidly relaxes, with about 10 per cent of the
particles moving inwards by a time of t = 5500, although with no
discernible further evolution over the time-frame of t = 5500–7500.
2 Note that the initial process of concentrating solid material into spiral
features described by Rice et al. (2004) – which is required for the formation
of planetesimals in the self-gravitating phase – instead involves the effect of
gas drag on small particles, and this, by definition, is ineffective in the test
particle regime considered here.
1908
J. Walmswell, C. Clarke and P. Cossins
We start by writing the angular momentum per unit mass, L, for
a particle at radius r as
r2
2
2
.
(5)
L = 2GM sin θ r −
2a
The equation is simplified by letting x = 1 − r/a, so that
L = GMa(1 − x 2 ) sin2 θ = L2max sin2 θ,
2
Figure 7. Evolution of the planetesimals initially beyond R = 10 at time
t = 5000.
The initial relaxation is consistent with the fact that the typical particle eccentricity is already ∼0.1 at t = 5000 and so just represents
the fact that particles near the ‘edge’ visit the inner region even
without further orbital evolution. We revisit the issue of planetesimal diffusion into an inner cleared region when we construct our
analytic model in Section 4.
4 A S I M P L E S C AT T E R I N G M O D E L
The simulation results encourage the exploration of a scenario in
which the evolution of the planetesimal swarm is primarily driven
by changes in angular momentum rather than energy. This prompted
us to attempt to reproduce our results by modelling the evolution of
the swarm as a process of diffusion in angular momentum space. We
start by obtaining analytic expressions for the diffusion coefficients,
set up the resulting diffusion equation and use Green’s functions to
solve for the evolution of the angular momentum distribution (and
the associated eccentricity and radial distributions) starting from
an arbitrary radial profile and initially circular orbits. The resulting solutions however involve the computation of large numbers of
Legendre polynomials and are not of practical use until the system
has evolved well away from the initial delta function distribution of
eccentricities. We nevertheless obtain solutions for the eccentricity
distribution that are a reasonable representation of the SPH results
at the end of the simulation. We can also compute the subsequent
evolution of the system and derive expressions for the final (equilibrium) distributions over radius and eccentricity (which correspond
to a uniform distribution in angular momentum at every energy).
In the following section, we then verify these predictions using a
simple Monte Carlo model; this approach has the additional advantage that it can be readily generalized to treat the case where the
perturbation amplitude is a function of radius.
4.1 Analytic solution
We consider an ensemble of particles orbiting a mass M with fixed
semimajor axis a. At some random time in each particle’s orbit the
velocity vector is rotated by some angle θ in the orbital plane,
where θ is the angle between the velocity and radial vectors. This
preserves the particle energy but perturbs the angular momentum.
(6)
and
dL 2
= Lmax − L2 .
(7)
dθ
Here, Lmax is the maximum local angular momentum and is a function of x (i.e. for particles at given r/a and energy – hence speed –
this maximum occurs when the velocity is purely tangential). The
angular momentum diffusivity, D, is given by L2 /τ where L is
the change in angular momentum associated with deflection θ and
τ is the time between such deflections: for now, we assume both
θ and τ to be constant around the orbit. We will also consider
the case where θ is independent of a and where τ scales with the
orbital period (∝a3/2 ). This is motivated by the wish to compare
with the SPH simulations which are conducted with constant cooling time to dynamical time-scale ratio and in which, therefore, the
fractional amplitude of gravitational disturbances is independent of
radius (see equation 3).
We can relate L to θ via equation (7) so that the value of D at
)2 , that is to say
given x is proportional to ( dL
dθ
2
dL
D∝
∝ GMa(1 − x 2 ) − L2 .
(8)
dθ
The expectation value of D over an entire orbit (at fixed L and a) is
obtained by averaging this quantity over time, using the relationship
between time interval and r for an elliptical orbit. Thus,
√
GMa 2
dr
=
e − (1 − r/a)2 ,
(9)
dt
r
where e is the orbital eccentricity, which is related to L and a by
L2 = GMa(1 − e2 ).
(10)
Equation (9) is derived from the orbital equation in the appendix.
As before, the equation is simplified by substituting x for r, so that
a 3/2 (1 − x)
dx.
dt ∝ (e2 − x 2 )
Thus, we have that (for fixed a)
+e
(GMa(1 − x 2 ) − L2 )(1 − x)
dx,
D ∝
(e2 − x 2 )
−e
(11)
(12)
where the odd terms integrate to zero over symmetric limits and
may be discarded. The remaining even terms are standard integrals
(requiring the substitution x = e sin w) and yield
1
(GMa − L2 ).
(13)
2
One can substitute this into Fick’s Laws of diffusion to get the
following 1D diffusion equation for the angular momentum distribution h(L, t). J is the flux and A(a) is a constant that is determined
by the strength of the perturbation and which in general depends
on a (see below). In the case that we are currently considering
(where the fractional amplitude of perturbations in the disc is independent of radius and where their characteristic time-scale τ scales
with the orbital period, i.e. ∝a3/2 ), then A is proportional to a−3/2 ;
D ∝
Planetesimals in self-gravitating discs
under these circumstances, the time-scale for angular momentum
diffusion scales with the local orbital period.
Thus, we have
J = −D
∂h
,
∂L
(14)
1909
owing to a prograde or retrograde orbit with angular momenta of
the same magnitude. The quantity dL is double-valued for the same
reason. Thus from equation (10) we have
L = ± GMa(1 − e2 ),
(24)
and
∂h
∂J
=−
,
∂t
∂L
(15)
D = A(a)(GMa − L2 ).
(16)
The form of D contrasts with other parametrizations in the
literature where it increases with L. Adams & Bloch (2009) consider,
for example, the case where it is proportional to L. The form of
equation (16) reflects the fact that where perturbations change only
the direction of the velocity, the associated angular momentum
change goes to zero in the limit of tangential orbits; this form thus
prevents diffusion of angular momentum beyond the physical limit
set by a circular orbit.
The diffusion equation is then
∂
∂h
2 ∂h
=
A(a)(GMa − L )
,
(17)
∂t
∂L
∂L
for which we assume a separable solution: h(L, t) = M(L)T(t) to
get
1 d
1 dT
2 dM
=
A(a)(GMa − L )
= −k .
(18)
T dt
M dL
dL
This gives T(t) ∝ exp(−kt). If k is non-negative, there will be no
growing solutions. The equation for M(L) is
k
d
2 dM
M = 0.
(19)
(GMa − L )
+
dL
dL
A(a)
√
Making the substitutions y = L/( GMa) and k/A(a) = n(n +
1) converts equation (19) into the Legendre equation:
d
2 dM
(1 − y )
+ n(n + 1)M = 0.
(20)
dy
dy
The solutions must be regular at y = ±1, so we may use the
Legendre polynomials. If this is combined with the solution for t,
we have
√
(21)
hn (L, t) = cn exp(−n(n + 1)A(a)t)Pn (L/ GMa).
√
∓e GMa
de.
dL = √
1 − e2
(25)
The Green function’s solution, f(e, t, a) for initially circular orbits
at a particular semimajor axis a is therefore
f (e, t, a) = √
Ne
1 − e2
∞
(2n + 1) exp(−n(n + 1)
n=0
× A(a)t)Pn (
1 − e2 ).
To consider the behaviour of an entire disc of particles, we then
calculate an appropriate superposition of the Green functions, bearing in mind (as argued above) that A = Ao a−3/2 . For comparison
with the SPH simulations, we take a number distribution per unit
a that is proportional to a−1/2 (since this corresponds to the initial
surface density distribution scaling as R−3/2 ) and adopt inner and
outer radii of a1 = 1 and a2 = 25. The integrations are straightforward but laborious, making Mathematica the obvious choice. We
plot F(e, t) (the distribution function for particle eccentricity) in
Fig. 8 and have normalized the time unit so as to match the peak
in the distribution obtained from the SPH simulation at a time t =
5000. At t = 0, F(e, t) is a delta function at e = 0 but this and the
subsequent evolution before t = 5000 required too many Legendre
polynomials to be computationally practicable. We however note
the similar form of the curve at t = 5000 to that obtained in the
SPH simulation at the same time (Fig. 7). The final distribution
occurs when t is large enough such that all the non-constant
Legen√
dre terms are effectively zero, leaving F (e, t) ∝ e/ 1 − e2 . This is
an equilibrium state and represents a uniform angular momentum
distribution for particles with the same semimajor axis.
The surface density distribution can be obtained through similar
means. Equation (11) gives the probability of a particle with eccentricity e being at x. Multiplying this by f(e, t, a) and integrating
with respect to e gives n(x, t, a), the fraction of particles at given
semimajor axis in a given interval of x. The lower limit is the value
The general solution is a linear combination of the hn solutions,
with the coefficients determined by the initial distribution:
h(L, t) =
∞
√
cn exp(−n(n + 1)A(a)t)Pn (L/ GMa).
(22)
n=0
If the initial distribution consists of N particles
√ in prograde circular orbits, we have h(L, t = 0) = N δ(L − GMa). If we use the
orthogonality property of the Legendre polynomials and the fact
that Pn (1) = 1 for all n, we have
h(L, t) = √
∞
√
2n + 1
exp(−n(n + 1)At)Pn (L/ GMa).
2
GMa n=0
N
(23)
The eccentricity distribution is then obtained by using f(e) de =
h(L) dL + h(−L) dL. This is because an eccentricity of e may be
(26)
Figure 8. Evolution of the normalized eccentricity distribution.
1910
J. Walmswell, C. Clarke and P. Cossins
of e for an orbit that achieves the required x value at either periapsis
or apoapsis:
1
1−x
f (e, t) √
de.
(27)
n(x, t, a) ∝
e2 − x 2
max(x,−x)
This must be converted to a function of r and a and then normalized so that it represents the distribution resulting from a fixed
number of particles. It can then be integrated with the power-law
distribution for a to get the overall number density:
a2
n(r, t, a)a −1/2 da,
(28)
N (r, t) ∝
a1
N (r, t)
.
(29)
2πr
Again, the integrations are complicated for very many Legendre
polynomials. However, the solution for the equilibrium distribution
is readily obtained. We note that (for a particle swarm of fixed
)−1 dr, which
a) the time spent in the interval r to r + dr is ( dr
dt
(using equations 9 and 5) can be written (for particles of angular
momentum L) as
(r, t) =
dt ∝ r dr
L2max − L2
.
(30)
Integrating over all values of L from −Lmax to Lmax , we obtain (in
general)
Lmax
h(L) dL
.
(31)
dt ∝ r dr
L2max − L2
−Lmax
In equilibrium, h is constant which means that one may write the
integral in terms of the dimensionless variable L̃ = L/Lmax so that
+1
dL̃
.
(32)
dt ∝ r dr
−1
1 − L̃2
We thus find that in equilibrium the fraction of particles between
r and r + dr scales as r dr. Now at fixed a, the minimum and
maximum radii attained by the particle are 0 and 2a so that the
normalization is such that
r dr
(r < 2a).
(33)
neq (r, a) =
2a 2
We can now find the surface density distribution by substituting
n(r, t, a) = ne q(r, a) into equations (28) and (29). Thus
1 a2 −5/2
N (r, t)
∝
(r, t) ∝
ra
da (a > r/2).
(34)
2πr
r a1
This gives rise to three regimes. For 0 < r < 2a1 , all values of
a contribute to the integral throughout this range and the surface
density is constant. For 2a1 < r < 2a2 then it is only a values greater
than r/2 which contribute; consequently, the surface density falls
with radius as the annulus at r is populated by an ever decreasing
range of a values. Finally, for r > 2a2 , no particles can be scattered
to this radius and the surface density is zero.
Summarizing, we therefore have
(r) = A (0 < r < 2a1 ),
(r) = A
(1 − (r/a2 )−3/2
(2a1 < r < 2a2 ),
1 − (a1 /a2 )−3/2
(r) = 0 (r > 2r2 ).
(35)
(36)
(37)
We note that although we have derived the above for a specific
distribution of particles in a, it is a general result that – in equilibrium
– the surface density is uniform for radii less than twice the inner
radius of the initial distribution. The reason for this is that every
group of particles of given a is redistributed, in equilibrium, into
a uniform density disc of particles extending from zero to 2a. At
every radial position inward of 2a1 , all bins of a provide uniform
surface density contributions to the total surface density. It is only
at radii >2a1 that particles with a < r/2 cease to contribute and the
surface density starts to decline.
We emphasize that this entire derivation is based on the assumption that the disc fluctuations are scale-free, which implies that θ
is independent of a and independent of phase at given a. In reality,
however, the amplitude of perturbations in self-gravitating discs is
an increasing function of radius. This changes the analysis in two
ways: it changes the dependence of D on L (cf. equation 13) because
the perturbations are of larger amplitude near apocentre and it also
changes the scaling of D with a. As an example, if (dθ )2 ∝ rp , then
the diffusion coefficient becomes (by analogy with equation 12)
+e
(GMa(1 − x 2 ) − L2 )(1 − x)p+1
dx.
(38)
D ∝ a p
(e2 − x 2 )
−e
In general, this integral is not analytic, except where p = −1: in
this case, the removal of the term (1 − x) makes no difference since
the term ∝x vanishes by symmetry for p = 0. Consequently, the
evolution at fixed a still reduces to the Legendre equation, although
the computation for a range of a now needs to take account of the
additional a dependence of D. We do not pursue this further here,
since – in considering situations that are likely to occur in real discs
– we want to be able to treat cases other than p = −1. This forms
part of our motivation for adopting a Monte Carlo method in the
following section.
4.2 Monte Carlo simulations
We have checked and extended the above analysis of the behaviour
of particle swarms that undergo diffusion in angular momentum at
fixed energy by conducting simple Monte Carlo simulations. Here,
we subject co-planar particles (initially in circular orbits with the
same surface density profile as employed in the SPH simulations)
to random rotation of their velocity vectors through an angle in the
range ±θ at a random time each orbit. As expected, the evolution
is qualitatively independent of the value of θ and with a time-scale
that scales as θ −2 . We have adjusted our parameters so as to match
the peak of the eccentricity distribution in the SPH calculation at
a time t = 5000 and verified that the evolution of the eccentricity
distribution (Fig. 9) is in good agreement with the output of our
analysis of the diffusion equation (Fig. 8). Secondly, the early evolution compares reasonably well with the f(e) curves obtained from
the SPH simulation (Fig. 4). However, the equilibrium distribution
for f(e) is only reached after several hundred thousand inner orbital
periods.
We also looked at the evolution of the surface density distribution
of the planetesimals with initial radii greater that R = 10 (Fig. 10),
so as to compare with that for the SPH simulation (Fig. 7). In the
Monte Carlo simulation, the surface density profile evolves through
a state similar to that seen in the SPH results and eventually attains
a true equilibrium, where the surface density is approximately flat
out to twice the inner radius (R = 20) and declines to zero at twice
the outer radius (R = 50). This is the result expected from the earlier
analysis.
Planetesimals in self-gravitating discs
1911
For 22 < R < 35 au (where opacity is dominated by ice sublimation) and
−9/14
R
β = 270
,
(40)
10 au
while for 4 < R < 22 au
−9/4
R
.
β = 1000
10 au
Figure 9. Evolution of the un-normalized eccentricity distribution from the
Monte Carlo simulation.
Figure 10. Evolution of the surface density distribution, in arbitrary units,
from the Monte Carlo simulation.
4.3 Application to real discs
We have hitherto adopted the assumption that the fractional amplitude of perturbations – and hence the value of θ – is independent
of radius. We have also normalized the evolution of the planetesimal swarm to that found in the SPH simulations which adopted
rather short cooling times (β = 5−10) and correspondingly large
amplitude of gravitational instability (with fractional amplitudes
of several tens of per cent; Cossins, Lodato & Clarke 2010). This
choice was motivated by the need to compute the evolution of the
planetesimal swarm in a reasonable time and also to ensure a sufficiently vigorous gravitational instability for it not to be quenched by
numerical viscosity (Lodato & Clarke 2011). The insights provided
by the simulations can now be used to rescale the problem to the
parameter range expected in real self-gravitating discs.
We therefore consider the case of an optically thick selfgravitating disc accreting at 3 × 10−6 M yr−1 for which the steady
state solutions (employing opacities due to dust and gas from Bell
& Lin 1994) are detailed in the appendix of Clarke (2009). At
radii >35 au, the opacity is dominated by ice and we have
−9/2
R
4
.
(39)
β = 3 × 10
10 au
(41)
Since the amplitude of perturbations scales as β −1/2 we also
expect that θ ∝ β −1/2 ; we normalize the value of θ at β = 5 in
order to match the rate of evolution of the SPH simulation. Within
R = 4 au, we set θ = 0 since it is arguable whether the disc is
self-gravitating at this point; in any case, the long cooling times
there equate to negligibly small values of θ .
We start with a belt of planetesimals in circular orbits which are
distributed with a power-law surface density distribution with ∝
R−3 since this is the steady-state gas surface density profile for a
self-gravitating disc with opacity dominated by ice grains (Clarke
2009). The belt extends from 60 to 100 au, the choice of inner radius being motivated by the fact that rather large amplitude density
enhancements (rapid cooling) are required in order to form planetesimals through dust concentration in spiral arms (Clarke & Lodato
2009). A primary motivation for modelling a distribution with an
initial hole is to discover whether – with a realistic prescription for
the radial dependence of the perturbation amplitude – one expects
planetesimals to be scattered inwards to fill up the hole over the
self-gravitating lifetime of the disc. The distributions of the eccentricity and the surface density are plotted in Fig. 11 and Fig. 12
respectively.
We find (Fig. 12) that the evolution is qualitatively similar to
that in Fig. 10: note that the edge is at 60 au in this simulation,
as opposed to 10 code units in Fig. 10. In both cases, the time is
normalized to the dynamical time at the inner edge (1 code unit
and 1 au, respectively); therefore, in order to compare the two plots
at a given number of orbital times at the truncation point, it is
necessary to multiply the times in Fig. 10 by 61.5 . It is then evident
that the time-scale for the infilling of the hole is rather similar in
the two cases, i.e. the rate of infill is controlled by the amplitude
of fluctuations near the truncation radius. This can be understood
inasmuch that particles that visit the inner disc on eccentric orbits
Figure 11. Evolution of the un-normalized eccentricity distribution from
the Monte Carlo simulation with realistic cooling.
1912
J. Walmswell, C. Clarke and P. Cossins
The picture that emerges from this study is that if planetesimals
do form in the self-gravitating phase of disc evolution, then they will
be retained in the disc throughout this phase; their high eccentricities inhibit collisions and mean that they will neither grow nor suffer
significant collisional disruption over this period. If (as argued by
Clarke & Lodato 2009) such planetesimals are only formed in the
outer disc, then they will be largely retained at such radii, though
with a significant minority that are perturbed to the inner regions of
the disc. Such an endpoint would therefore represent the initial conditions for considering the subsequent evolution of planetesimals
during the non-self-gravitating phase of disc evolution.
AC K N OW L E D G E M E N T S
Figure 12. Evolution of the surface density distribution, in arbitrary units,
from the Monte Carlo simulation with realistic cooling.
spend most of their time near apocentre (i.e. close to the truncation
radius). The fact that – in the realistic variable β case – the spiral
structure is of very low amplitude in the inner disc therefore has
little effect on the evolution of the planetesimal swarm.
5 CONCLUSIONS
The SPH simulations indicate that planetesimal eccentricity is
efficiently amplified by interaction with spiral features in selfgravitating discs. In the case of discs where the cooling time is
not much longer than the dynamical time-scale (i.e. in the outer
disc, at radii >10 s of au, where we expect planetesimals to be
formed in such discs), the amplitude of these features is sufficient
for high eccentricities (>0.1) to be driven on much less than the
self-gravitating lifetime of the disc. We have found that there is
no significant velocity coherence within the particle swarm and
thus that the local velocity dispersion is a significant fraction of
the local orbital velocity; we also find that there is no tendency –
in the test particle regime studied here – for the planetesimals to
be concentrated within spiral arms. The lack of significant density
enhancements and the large velocity dispersion (which weakens
gravitational focusing) both contribute to very long collision times
(> a Gyr); moreover, any collisions that did occur would be in the
destructive regime.
We have used the SPH simulations to calibrate Monte Carlo experiments in which the particle direction is randomly perturbed
around its orbit and have compared these Monte Carlo experiments
with an analytical description. The Monte Carlo experiments allow
the long time integration of the system and can also probe the high
cooling time (weak spiral) regime that cannot be reliably simulated
hydrodynamically. We have treated the case of an initial planetesimal belt at large radius (60 au) where the radial variation of the
spiral amplitude is parametrized in terms of expected variations
in the local cooling physics in marginally unstable self-gravitating
discs. We find that – notwithstanding the fact that the spiral potential is very weak at small radius – there is significant scattering
of planetesimals into the inner disc, with particles that are perturbed in the region beyond 60 au visiting small radii on eccentric
orbits. We nevertheless find that most of the particles initially beyond 60 au are likely to be retained at large radius on time-scales of
∼105 yr.
JJW thanks the STFC for his studentship. We would like to thank
Guiseppe Lodato for providing much valuable advice, Mark Wyatt
and Jim Pringle for discussion and comments, and John Eldridge
for proofreading the paper.
REFERENCES
Adachi I., Hayashi C., Nakazawa K., 1976, Prog. Theor. Phys., 56, 1756
Adams F. C., Bloch A. M., 2009, ApJ, 701, 1381
Bate M. R., Bonnell I. A., Price N. M., 1995, MNRAS, 277, 362
Bate M. R., Lubow S. H., Ogilvie G. I., Miller K. A., 2003, MNRAS, 341,
213
Bell K. R., Lin D. N. C., 1994, ApJ, 427, 987
Benz W., Asphaug E., 1999, Icarus, 142, 5
Binney J., Tremaine S., 2008, Galactic Dynamics, 2nd edn. Princeton Univ.
Press, Princeton, NJ
Boss A. P., 2000, ApJ, 536, L101
Britsch M., Clarke C. J., Lodato G., 2008, MNRAS, 385, 1067
Clarke C. J., 2009, MNRAS, 396, 1066
Clarke C. J., Lodato G., 2009, MNRAS, 398, L6
Cossins P., Lodato G., Clarke C. J., 2009, MNRAS, 393, 1157
Cossins P., Lodato G., Clarke C., 2010, MNRAS, 401, 2587
Gammie C. F., 2001, ApJ, 553, 174
Gibbons P. G., Rice W. K. M., Mamatsashvili G. R., 2012, MNRAS, 426,
1444
Greaves J. S., Richards A. M. S., Rice W. K. M., Muxlow T. W. B., 2008,
MNRAS, 391, L74
Hubickyj O., Bodenheimer P., Lissauer J. J., 2005, Icarus, 179, 415
Ida S., Guillot T., Morbidelli A., 2008, ApJ, 686, 1292
Kokubo E., Ida S., 2000, Icarus, 143, 15
Laughlin G., Bodenheimer P., Adams F. C., 2004, ApJ, 612, L73
Leinhardt Z. M., Stewart S. T., Schultz P. H., 2008, in Barucci M. A.,
Boehnhardt H., Cruikshank D. P., Morbidelli A., Dotson R., eds, Physical
Effects of Collisions in the Kuiper Belt, Univ. of Arizona Press, Tuscon,
p. 195
Lodato G., Clarke C. J., 2011, MNRAS, 413, 2735
Lodato G., Rice W. K. M., 2005, MNRAS, 358, 1489
Meru F., Bate M. R., 2011, MNRAS, 410, 559
Meru F., Bate M. R., 2012, MNRAS, 427, 2022
Nelson R. P., 2005, A&A, 443, 1067
Nelson R. P., Gressel O., 2010, MNRAS, 409, 639
Nelson R. P., Papaloizou J. C. B., 2004, MNRAS, 350, 849
Paardekooper S.-J., 2012, MNRAS, 421, 3286
Paardekooper S.-J., Baruteau C., Meru F., 2011, MNRAS, 416, L65
Pollack J. B., Hubickyj O., Bodenheimer P., Lissauer J. J., Podolak M.,
Greenzweig Y., 1996, Icarus, 124, 62
Rice W. K. M., Lodato G., Pringle J. E., Armitage P. J., Bonnell I. A., 2004,
MNRAS, 355, 543
Rice W. K. M., Lodato G., Pringle J. E., Armitage P. J., Bonnell I. A., 2006,
MNRAS, 372, L9
Rice W. K. M., Forgan D. H., Armitage P. J., 2012, MNRAS, 420, 1640
Planetesimals in self-gravitating discs
Safronov V. S., 1969, Evoliutsiia doplanetnogo oblaka. Nauka, Moscow
Stamatellos D., Whitworth A. P., Bisbas T., Goodwin S., 2007, A&A, 475,
37
Takeuchi T., Clarke C. J., Lin D. N. C., 2005, ApJ, 627, 286
Tanaka H., Ward W. R., 2004, ApJ, 602, 388
Toomre A., 1964, ApJ, 139, 1217
Wyatt M. C., Smith R., Greaves J. S., Beichman C. A., Bryden G., Lisse
C. M., 2007, ApJ, 658, 569
A P P E N D I X A : I N T E G R AT I N G A RO U N D
AN ORBIT
The expected value of D is obtained by averaging over one orbit.
This requires obtaining dt as a function of r. We start with the orbital
equation for r as a function of φ:
r=
a(1 − e2 )
1 + e cos φ
dr dφ
dr L
dr
=
=
.
dt
dφ dt
dφ r 2
1913
Here, L is a constant of the motion for given values of a and e:
L2 = GMa(1 − e2 )
dr
a(1 − e2 )e sin φ
=
dt
(1 + e cos φ)2
(A3)
GMa(1 − e2 )
.
r2
Substituting r back in simplifies the expression:
√
GMa sin φ
dr
=
dt
1 − e2 a
(A4)
(A5)
Re-arranging equation (A1) and using that to eliminate φ gives
an expression in r and e only:
√
GMa 2
dr
e − (1 − r/a)2 .
(A6)
=
dt
r
(A1)
(A2)
This paper has been typeset from a TEX/LATEX file prepared by the author.